Charge Orbits of Symmetric Special Geometries and Attractors

TL;DR

The paper classifies regular attractor solutions for N=2, d=4 symmetric MESGTs, showing three distinct attractor types (1/2-BPS, non-BPS with Z≠0, and non-BPS with Z=0) that correspond to non-degenerate charge orbits under the U-duality group and are distinguished by the quartic invariant I4. Using Freudenthal triple systems and Jordan algebras, the authors connect the orbit structure to the scalar mass spectra via symmetry stabilizers, providing explicit decompositions for magical theories based on J3^O and J3^H and their N=6 duals. The mass spectra at critical points follow a pattern where 1/2-BPS are strictly stable, non-BPS Z≠0 split into massive/massless modes aligned with hat{h}, and non-BPS Z=0 align with tilde{h}′; these results are illustrated in the stu-like models and the exceptional cases. The work highlights a deep interplay between four- and five-dimensional duality groups, the geometry of the scalar manifolds, and the attractor mechanism, offering a framework for extending to non-symmetric and non-cubic cases. It also reveals a cross-symmetry between N=2 and N=6 theories via the shared structure of the SO^*(12)/U(6) sector and its charge orbits.

Abstract

We study the critical points of the black hole scalar potential $V_{BH}$ in N=2, d=4 supergravity coupled to $n_{V}$ vector multiplets, in an asymptotically flat extremal black hole background described by a 2(n_{V}+1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold. For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with non-vanishing Bekenstein-Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2(n_{V}+1)-dimensional representation $R_{V}$ of the U-duality group. Such orbits are non-degenerate, namely they have non-vanishing quartic invariant (for rank-3 spaces). Other than the 1/2-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge. The three species of solutions to the N=2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of $V_{BH}$ and some group theoretical considerations on homogeneous symmetric special Kähler geometry.